Page 415 - Euclid's Elements of Geometry

ST EW iþ. riaþ ELEMENTS BOOK 10

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An apotome is not the same as a binomial.
῾Η ἀποτομὴ οὐκ ἔστιν ἡ αὐτὴ τῇ ἐκ δύο ὀνομάτων. show. Proposition 111
Α Β A B

∆ Η Ε Ζ D G E F

Γ C
῎Εστω ἀποτομὴ ἡ ΑΒ· λέγω, ὅτι ἡ ΑΒ οὐκ ἔστιν ἡ αὐτὴ Let AB be an apotome. I say that AB is not the same
τῇ ἐκ δύο ὀνομάτων. as a binomial.
Εἰ γὰρ δυνατόν, ἔστω· καὶ ἐκκείσθω ῥητὴ ἡ ΔΓ, καὶ τῷ For, if possible, let it be (the same). And let a rational
ἀπὸ τῆς ΑΒ ἴσον παρὰ τὴν ΓΔ παραβεβλήσθω ὀρθογώνιον (straight-line) DC be laid down. And let the rectangle
τὸ ΓΕ πλάτος ποιοῦν τὴν ΔΕ. ἐπεὶ οὖν ἀποτομή ἐστιν ἡ ΑΒ, CE, equal to the (square) on AB, have been applied to
ἀποτομὴ πρώτη ἐστὶν ἡ ΔΕ. ἔστω αὐτῇ προσαρμόζουσα ἡ CD, producing DE as breadth. Therefore, since AB is an
ΕΖ· αἱ ΔΖ, ΖΕ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι, καὶ apotome, DE is a first apotome [Prop. 10.97]. Let EF
ἡ ΔΖ τῆς ΖΕ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, καὶ ἡ be an attachment to it. Thus, DF and FE are rational
ΔΖ σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΔΓ. πάλιν, (straight-lines which are) commensurable in square only,
ἐπεὶ ἐκ δύο ὀνομάτων ἐστὶν ἡ ΑΒ, ἐκ δύο ἄρα ὀνομάτων and the square on DF is greater than (the square on) FE
πρώτη ἐστὶν ἡ ΔΕ. διῃρήσθω εἰς τὰ ὀνόματα κατὰ τὸ Η, by the (square) on (some straight-line) commensurable
καὶ ἔστω μεῖζον ὄνομα τὸ ΔΗ· αἱ ΔΗ, ΗΕ ἄρα ῥηταί εἰσι (in length) with (DF), and DF is commensurable in
δυνάμει μόνον σύμμετροι, καὶ ἡ ΔΗ τῆς ΗΕ μεῖζον δύναται length with the (previously) laid down rational (straight-
τῷ ἀπὸ συμμέτρου ἑαυτῇ, καὶ τὸ μεῖζον ἡ ΔΗ σύμμετρός line) DC [Def. 10.10]. Again, since AB is a binomial,
ἐστι τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΔΓ. καὶ ἡ ΔΖ ἄρα τῇ ΔΗ DE is thus a first binomial [Prop. 10.60]. Let (DE) have
σύμμετρός ἐστι μήκει· καὶ λοιπὴ ἄρα ἡ ΗΖ σύμμετρός ἐστι been divided into its (component) terms at G, and let
τῇ ΔΖ μήκει. [ἐπεὶ οῦν σύμμετρός ἐστιν ἡ ΔΖ τῇ ΗΖ, ῥητὴ DG be the greater term. Thus, DG and GE are rational
δέ ἐστιν ἡ ΔΖ, ῥητὴ ἄρα ἐστὶ καὶ ἡ ΗΖ. ἐπεὶ οὖν σύμμετρός (straight-lines which are) commensurable in square only,
ἐστιν ἡ ΔΖ τῇ ΗΖ μήκει] ἀσύμμετρος δὲ ἡ ΔΖ τῇ ΕΖ μήκει. and the square on DG is greater than (the square on)
ἀσύμμετρος ἄρα ἐστὶ καὶ ἡ ΖΗ τῇ ΕΖ μήκει. αἱ ΗΖ, ΖΕ ἄρα GE by the (square) on (some straight-line) commensu-
ῥηταί [εἰσι] δυνάμει μόνον σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ rable (in length) with (DG), and the greater (term) DG
ΕΗ. ἀλλὰ καὶ ῥητή· ὅπερ ἐστὶν ἀδύνατον. is commensurable in length with the (previously) laid
῾Η ἄρα ἀποτομὴ οὐκ ἔστιν ἡ αὐτὴ τῇ ἐκ δύο ὀνομάτων· down rational (straight-line) DC [Def. 10.5]. Thus, DF
ὅπερ ἔδει δεῖξαι. is also commensurable in length with DG [Prop. 10.12].
The remainder GF is thus commensurable in length with
DF [Prop. 10.15]. [Therefore, since DF is commensu-
rable with GF, and DF is rational, GF is thus also ra-
tional. Therefore, since DF is commensurable in length
with GF,] DF (is) incommensurable in length with EF.
Thus, FG is also incommensurable in length with EF
[Prop. 10.13]. GF and FE [are] thus rational (straight-
lines which are) commensurable in square only. Thus,

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